Fully packed loop configurations: polynomiality and nested arches

This article proves a conjecture by Zuber about the enumeration of fully packed loops (FPLs). The conjecture states that the number of FPLs whose link pattern consists of two noncrossing matchings which are separated by $m$ nested arches is a polynomial function in $m$ of certain degree and with certain leading coefficient. Contrary to the approach of Caselli, Krattenthaler, Lass and Nadeau (who proved a partial result) we make use of the theory of wheel polynomials developed by Di Francesco, Fonseca and Zinn-Justin. We present a new basis for the vector space of wheel polynomials and a polynomiality theorem in a more general setting. This allows us to finish the proof of Zubers conjecture.


Introduction
Razumov and Stroganov conjectured in [5] a relation between fully packed loop configurations (short FPLs) and the ground state vector in the O(1) loop model. This conjectural connection made it possible to come up with many new conjectures concerning FPLs with certain link patterns, see e. g. [11]. In this paper we will prove the following Theorem 1.1 ([11,Conjecture 7]) For noncrossing matchings π ∈ N C n and π ∈ N C n and an integer m the number of FPLs with link pattern (π) m π is a polynomial in m of degree |λ(π)| + |λ(π )| with leading coefficient dim(λ(π)) dim(λ(π )) |λ(π)|!|λ(π )|! . The special case for only one matching was completely shown in [2][ Theorem 4.2], whereas the general case was proven only for large values of m, see [2][Theorem 6.7]. With the proof of the Razumov-Stroganov Conjecture in [1], it was possible to use new methods -namely the theory of wheel polynomials which was developed in [2], [9] -for solving open problems concerning FPLs. Using integral wheel polynomials, Fonseca and Zinn-Justin reproved in [4] Theorem 1.1 for the special case of one matching. By considering another well known family of wheel polynomials we will be able to prove Theorem 1.1.

Noncrossing matchings
A noncrossing matching π of size n is a derangement π ∈ S 2n with π 2 = Id 2n such that there exist no integers 1 ≤ a < b < c < d ≤ 2n with π(a) = c and π(b) = d. We denote by N C n the set of noncrossing matchings of size n. The number of noncrossing matchings of size n is given by the n-th Catalan number C n = 1 n+1 2n n . A noncrossing matching π can be represented graphically by 2n dots which are labeled from 1 to 2n and two dots are linked by an arc iff their labels are mapped onto each other by π, see Figure  1. In the following we do not want to distinguish between the graphical representation of a noncrossing matching π and π itself.
A Young diagram is a finite collection of boxes, arranged in left-justified rows and weakly decreasing row-length from top to bottom (English notation). The noncrossing matchings of size n are in bijection with the Young diagrams for which the i-th row from the top consists of at most n − i boxes. We construct for a noncrossing matching π its corresponding Young diagram λ(π) by reading its parenthesis word from left to right and make an up-step if an opening bracket is read and a right-step if a closing bracket is read. The Young diagram λ(π) is then the area enclosed by this path and the path taking n consecutive up-steps followed by n consecutive right-steps, see Figure 2. To keep in mind which size the noncrossing matching π with corresponding Young diagram λ(π) has and for reasons which will be clear later, we will always draw these n consecutive up-/ right-steps we needed in the above construction when the Young diagram is depicted Since Young diagrams are in bijection with integer partitions, noncrossing matchings of size n are in bijection to integer partitions where the i-th summand is less or equal than n − i. The bijection between Young diagrams and integer partitions is in this work given by setting the i-th summand of a partition equal to the number of boxes of the i-th row from the top of a partition. For a Young diagram λ we denote with |λ| the number of its boxes.
Let π, σ be noncrossing matchings of size n or n respectively and m a positive integer. We denote by (π) m the noncrossing matching of size n + m which is obtained by putting m nested arches around π and by () m the noncrossing matching which is made out of m consecutive small arches. Further we denote by πσ the concatenation of π and σ, see Figure 3.  Let k be the index such that π(k) = 2n. Then define ρ(π) as the rotation of π given by ρ(π)(i) = π(i − 1) + 1 for 1 < i ≤ 2n and ρ(π)(1) = k + 1. For 1 ≤ j ≤ 2n define e j : N C n → N C n as the map where j + 1 is replaced by 1 if j = 2n. It is easy to show that ρ(π) and e j (π) for 1 ≤ j ≤ 2n are again noncrossing matchings of size n. For a graphical representation of e j see Figure 4.
We can define a partial order on the set N C n of all noncrossing matchings of size n by defining σ ≤ π for two elements σ, π ∈ N C n if the Young diagram λ(σ) is contained in λ(π). For 2 ≤ j ≤ 2n − 2 we denote σ j π if λ(π) is obtained from λ(σ) by adding one box with top left corner on the j-th south-east diagonal, where these diagonals are counted as in Figure 5. This notion is obviously a refinement of the above defined partial order.

Fully packed loop configurations
A fully packed loop configuration (or short FPL) F of size n is a subgraph of the n × n grid with n external edges on every side with the following two properties: 1. F contains all vertices of the n × n grid and every vertex of F has degree 2.
2. F contains every other external edge, beginning with the topmost at the left side.
We enumerate the external edges in a FPL counter-clockwise with 1 up to 2n, see Figure 6. A FPL consists out of paths which do not cross each other, the closed ones are called Loops. We can assign to every FPL F a noncrossing matching π(F ) by setting π(F )(i) the label of the external edge which is connected to the i-th external edge for 1 ≤ i ≤ 2n, and call π(F ) the link pattern of F .
It is well known that there exists a bijection between all FPLs of size n and all Alternating Sign Matrices (or short ASMs) of size n, see for example [11]. Hence the number A n of all FPLs of size n is equals to the number A n of ASMs of size n, see [8] Let π be a noncrossing matching of size n, we denote with A π the number of all FPLs of size n with link pattern π. In [7] it was shown that the number of FPLs with link pattern π is invariant under the action of the dihedral group D n onto the noncrossing matchings of size n.

Wheel polynomials
We will introduce in this section wheel polynomials only succinctly, for a more detailed overview see for example [6,Section 2]. For a positive integer n a polynomial p ∈ Q(q)[z 1 , . . . , z 2n ] is called wheel polynomial of order n if p is a homogeneous polynomial of degree n(n − 1) and satisfies the wheel condition: the space of all wheel polynomials of order n with coefficients in Q(q).
It is not difficult to show the following of all wheel polynomials of order n is closed under the action of D k for k ∈ {1, . . . , 2n}.
2. For all k ∈ {1, . . . , 2n} and all polynomials f, g ∈ Q(q)[z 1 , . . . , z 2n ] one has: This product rule can also be stated more generally for an integer m and The next theorem describes a Q(q)-basis of the space W n [z] of all wheel polynomials of order n and a recursion to calculate them. However the most important result for us is that we can calculate the number of FPLs with a certain noncrossing matching as link pattern by evaluating the proper element of this basis at z i = 1 for all 1 ≤ i ≤ 2n. The proofs for the statements can be found in [9,Section 4], whereas the proof of the last statement also requires the Razumov-Stroganov Conjecture, which was proven in [1].
indexed by the noncrossing matchings of size n with the following properties: a root of unity, then Ψ π (1, . . . , 1) = A π holds for all π ∈ N C n .
It is easy to prove that the set of preimages of the noncrossing matching σ under e j as in the above setting is a subset of {σ, π} ∪ {τ ∈ N C n : τ < σ}. Since noncrossing matchings and Dyck paths are in bijection we can define for all 1 ≤ j ≤ 2n a map corresponding to e j for Dyck paths. With the explicit description of these maps in [3, Proposition 2.1] it is easy to prove the following Lemma 2.4 Let α, β be noncrossing matchings of size n or n respectively and m an integer. Then Remark 2.5 Let π be a noncrossing matching of size n and denote with Λ π the set of indices such that all boxes of λ(π) lie on diagonals which are indexed with numbers in Λ π . One can show inductively using the first two statements of the Theorem 2.3, that Ψ π is a sum of terms of the form where m ≤ |λ(π)| and i 1 , · · · i m ∈ Λ π . The number of summands and the indices i 1 , . . . , i m appearing in the summands depend only on shape of the Young diagram λ(π). This implies that for any positive integer m the wheel polynomial Ψ (π)m has the same presentation as a sum of the above form as Ψ π except that Ψ ()n is replaced by Ψ ()n+m in the summands. Hence this presentation of Ψ (π)m as a sum depends only on π and therefore independent on m.

The proof
The following notations are only for the sake of brevity: One verifies by simple calculation: Let m be a positive integer. Then the following holds: We use the following notation: The following example is included to sketch the main idea of the proof of the above Theorem and to help the reader in understanding it. Example 3.3 Let n = m = i m = 1 Then we can calculate D 1 (P ) z1=z2=1 explicitly, where m, n, i m and P are as above. By using Lemma 2.2 and Lemma 3.1 we obtain for D 1 (P ) the following expression: If we evaluate this for z 1 = z 2 = 1 we obtain: Proof of Theorem 3.2: We prove the theorem by induction on m. The statement is trivial for m = 0, hence let m > 0 and set k := i m . We can write for a finite set S of indices, a s ∈ {±1, ±q, ±q −1 } and P s = P (α i,j;s |β i;s |γ i;s ) for all s ∈ S. Indeed we can use iteratively the second statement of Lemma 2.2 to split the product into a sum. This splitting depends on the order of the factors of the product, the order we will apply is the one which will be used for the calculations in (1) and was used in Example 3.3. Lemma 3.1 implies that every summand is of the form P s = P (α i,j;s |β i;s |γ i;s ) and the coefficients a s are as stated above. Therefore we need to calculate For this calculation we split S into subsets A i,j , B i and C i with 1 ≤ i = j ≤ 2n by using Lemma 2.2 in the following way: We define A i,j ⊂ S to be the set of all s ∈ S such that the summands P s originate from calculating the summand in Equation 1 with the factor D k (f (i, j) αi,j ) explicitly -this can be done by using Lemma 3.1. The sets B i and C i are defined analogously where the role of f (i, j) αi,j is replaced by g(i) βi or h(i) γi respectively. Hence we can write the set S as the union Lemma 3.1 implies D k (f (i, j)) = 0 for {i, j} ∩ {k, k + 1} = ∅ and D k (g(i)) = D k (h(i)) = 0 for i / ∈ {k, k + 1}. Hence the sets A i,j , B i , C i are empty in these cases.
Then the definition of A i,j and Lemma 3.1 implies for all (i , j ) / ∈ {(i, j), (σ(i), σ(j)), (k, k + 1)} and all s ∈ A i,j : For (k, k + 1) / ∈ {(i, j), (σ(i), σ(j))} one obtains the parameter α k,k+1;s by adding one to the value of α i ,j in the above description for (i , j ) = (k, k + 1).Further we obtain β i ;s = β σ(i ) and γ i ;s = γ σ(i ) for all 1 ≤ i ≤ 2n and s ∈ A i,j . By Lemma 3.1 the constant a s is for all s ∈ A i,j determined by the corresponding constant of D k (f (i, j)) and hence not depending on s. The last statement of Lemma 3.1 implies that we can list the elements of A i,j = {s 1 , . . . , s αi,j } such that we have the following description for the remaining parameters α i,j;s , α σ(i),σ(j);s : For the sets B i , C i and i ∈ {k, k + 1} we have an analogue description as above, whereas the only parameters depending on s ∈ B i or s ∈ C i respectively are given in the case of B i by with 1 ≤ t ≤ β i and in the case of C i by By induction we know that D i1 • · · · • D im−1 (P (α i,j;s |β i;s |γ i;s )) |z1=...=z2n=1 is a polynomial Q of degree at most m − 1 in (α i,j;s ), (β i;s ), (γ i;s ) for all s ∈ S. The above description implies that if we restrict ourself to s ∈ A i,j , s ∈ B i or s ∈ C i respectively, a s is invariant in s, the parameters (α i,j;s ), (β i;s ), (γ i;s ) are linear combinations of (α i,j ), (β i ), (γ i ) and in two cases also depending linearly on a parameter t which runs from 1 up to the cardinality of the set A i,j , B i or C i respectively. Hence the sum s∈Ai,j a s D i1 • · · · • D im−1 (P s ) |z1=···=z2n=1 , and the analogous sums for s ∈ B i or s ∈ C i respectively are polynomials in (α i,j ), (β i ), (γ i ) of degree at most m. Therefore Proof of Theorem 1.1: In the following we will show that the integer A (π)mπ of FPLs with link pattern (π) m π is a polynomial in m. Together with [2][Theorem 6.7], which states that A (π)mπ is a polynomial in m with requested degree and leading coefficient for large values of m, this proves the theorem.
Denote the corresponding partition of π by (λ 1 , · · · , λ p ) and let q = e 2πi 3 throughout the proof. By Theorem 2.3, the number A (π)mπ is given by We calculate Ψ (π)mπ in the following three steps: 1. First we calculate Ψ (π) m+n . The Young diagram corresponding to (π) m+n consists of |λ(π)| boxes which lie on the south-east-diagonals which are indexed by the numbers m + n + 2, . . . , m + 2n + n − 2. Hence by the recursion in Theorem 2.3 the wheel polynomial Ψ (π) m+n is a sum of terms of the form D i1 • · · · • D i k Ψ () m+n+n -as explained in Remark 2.5 -over k ≤ |λ(π)| and i 1 , . . . , i k ∈ {m + n + 2, . . . , m + 2n + n − 2}. As mentioned in Remark 2.5, the number of summands and the indices i j of the linear maps D ij in the summands depend only on π and not on m.
By Theorem 3.2 the evaluation z i = 1 for the remaining variables in the above expressions are polynomials in (α i,j ), (β i ), (γ i ) of degree at most k ≤ |λ(π)| + |λ(π )|. The explicit choice of (α i,j ), (β i ), (γ i ) implies that it is actually a polynomial in m of degree at most |λ(π)| + |λ(π )|. Since A (π)mπ is a sum of these polynomials and the sum is independent of m this proves the theorem. 2 In the above proof we used the fact that in [2] it was already shown that the number A (π)mπ is a polynomial in m with the correct degree and leading coefficient for large values of m. However it seems to be possible to prove the statement about the degree and the leading coefficient by means of wheel polynomials -this is work in progress.